Gökova Geometry Topology Institute

Time: 4:00 PM IST
Title: The homogeneous flow and the Poincare Conjecture
Abstract: Recently we introduced a theory of geometric structures (called prehomogeneous geometries PHGs) parametrized by some integer m, called the order of the PHG. The curvature of the PHG measures the deviation of the PHG from some fully symmetric model. For instance, a Riemannian PHG arises for m=1 and in this case there are 3 fully symmetric models: The curvature vanishes if and only if the metric is of constant curvature. For m=0, the vanishing of the curvature of an absolute parallelism gives rise to the theory of Lie groups and Lie algebras. In this talk, we will introduce the main concepts of this theory in the case m=0 in some detail. If time permits, we will outline the construction of a flow which starts with an arbitrary absolute parallelism as initial condition and flows in the direction of an absolute parallelism whose curvature vanishes. This flow, like the Ricci flow, is a second order nonlinear parabolic evolution equation and is much more elementary than the Ricci flow.